A336614 Number of n X n (0,1)-matrices A over the reals such that A^2 is the transpose of A.

1, 2, 4, 10, 32, 112, 424, 1808, 8320, 40384, 210944, 1170688, 6783616, 41411840, 265451008, 1765520128, 12227526656, 88163295232, 656548065280, 5054719287296, 40261285543936, 330010835894272, 2783003772452864, 24166721466204160, 215318925894909952, 1966855934183800832

Offset

0,2

Comments

From Peter Luschny, Jun 04 2021: (Start)

a(n) = n! * [x^n] exp(x*(x^2 + 6)/3).

a(n) = 2*a(n - 1) + (n^2 - 3*n + 2)*a(n - 3) for n >= 3.

a(n) = Sum_{k=0..n/3} (2^(n-3*k)*n!)/(3^k*k!*(n-3*k)!).

a(n) = 2^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [], -9/8).

[The above formulas, first stated as conjectures, were proved by mjqxxxx at Mathematics Stack Exchange, see link.] (End)

Links

Table of n, a(n) for n=0..25.

mjqxxxx, Proof of conjectured formulas, Mathematics Stack Exchange.

Formula

a(n) = A336174(n) + A000079(n).

Example

a(3) = A336174(3) + A000079(3) = 2 + 8 = 10.

Maple

a := n -> add((2^(n - 3*k)*n!)/(3^k*k!*(n - 3*k)!), k=0..n/3):
seq(a(n), n=0..25); # Peter Luschny, Jun 05 2021

Prog

(PARI) m(n, t) = matrix(n, n, i, j, (t>>(i*n+j-n-1))%2)
a(n) = sum(t = 0, 2^n^2-1, m(n, t)^2 == m(n, t)~)
for(n = 0, 9, print1(a(n), ", "))
(Python)
from itertools import product
from sympy import Matrix
def A336614(n):
    c = 0
    for d in product((0, 1), repeat=n*n):
        M = Matrix(d).reshape(n, n)
        if M*M == M.T:
            c += 1
    return c # Chai Wah Wu, Sep 29 2020

Crossrefs

Row sums of A344912.

Cf. A000079, A001471, A336174.

Sequence in context: A363138 A367113 A365516 * A071954 A352279 A120017

Adjacent sequences: A336611 A336612 A336613 * A336615 A336616 A336617

Keyword

nonn

Author

Torlach Rush, Jul 27 2020

Extensions

More terms from Peter Luschny, Jun 05 2021

Status

approved